International Mathematical Olympiad 1993 Problem 4

For three points $P, Q, R$ in the plane, we define $m(PQR)$ as the minimum length of the three altitudes of $\triangle PQR$. (If the points are collinear, we set $m(PQR) = 0$.)

Prove that for points $A, B, C, X$ in the plane, $$m(ABC) \leq m(ABX) + m(AXC) + m(XBC).$$